fourierdlem25 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem25 46581

Description: If 𝐶 is not in the range of the partition, then it is in an open interval induced by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem25.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem25.qf	⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
fourierdlem25.cel	⊢ (𝜑 → 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
fourierdlem25.cnel	⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝑄)
fourierdlem25.i	⊢ 𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < )

**Assertion**
Ref	Expression
fourierdlem25	⊢ (𝜑 → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))))

Distinct variable groups: 𝐶,𝑘 𝐶,𝑗 𝑗,𝐼 𝑘,𝐼 𝑘,𝑀 𝑗,𝑀 𝑄,𝑘 𝑄,𝑗 Allowed substitution hints: 𝜑(𝑗,𝑘)

Proof of Theorem fourierdlem25 Dummy variables ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem25.i	. . 3 ⊢ 𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < )
2		ssrab2 4021	. . . 4 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ (0..^𝑀)
3		ltso 11220	. . . . . 6 ⊢ < Or ℝ
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → < Or ℝ)
5		fzofi 13930	. . . . . . 7 ⊢ (0..^𝑀) ∈ Fin
6		ssfi 9101	. . . . . . 7 ⊢ (((0..^𝑀) ∈ Fin ∧ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ (0..^𝑀)) → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin)
7	5, 2, 6	mp2an 693	. . . . . 6 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin)
9		0zd 12530	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℤ)
10		fourierdlem25.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
11	10	nnzd 12544	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12	10	nngt0d 12220	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝑀)
13		fzolb 13614	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑀) ↔ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
14	9, 11, 12, 13	syl3anbrc 1345	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
15		fourierdlem25.qf	. . . . . . . . 9 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
16		elfzofz 13624	. . . . . . . . . 10 ⊢ (0 ∈ (0..^𝑀) → 0 ∈ (0...𝑀))
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...𝑀))
18	15, 17	ffvelcdmd 7032	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
19	10	nnnn0d 12492	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
20		nn0uz 12820	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
21	19, 20	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
22		eluzfz2 13480	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
24	15, 23	ffvelcdmd 7032	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
25	18, 24	iccssred 13381	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄‘0)[,](𝑄‘𝑀)) ⊆ ℝ)
26		fourierdlem25.cel	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
27	25, 26	sseldd 3923	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
28	18	rexrd 11189	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ^*)
29	24	rexrd 11189	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ^*)
30		iccgelb 13349	. . . . . . . . 9 ⊢ (((𝑄‘0) ∈ ℝ^* ∧ (𝑄‘𝑀) ∈ ℝ^* ∧ 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → (𝑄‘0) ≤ 𝐶)
31	28, 29, 26, 30	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑄‘0) ≤ 𝐶)
32		fourierdlem25.cnel	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝑄)
33		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 𝐶 = (𝑄‘0))
34	15	ffnd 6664	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
35	34	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 𝑄 Fn (0...𝑀))
36	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 0 ∈ (0...𝑀))
37		fnfvelrn 7027	. . . . . . . . . . . 12 ⊢ ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) → (𝑄‘0) ∈ ran 𝑄)
38	35, 36, 37	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → (𝑄‘0) ∈ ran 𝑄)
39	33, 38	eqeltrd 2837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = (𝑄‘0)) → 𝐶 ∈ ran 𝑄)
40	32, 39	mtand 816	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝐶 = (𝑄‘0))
41	40	neqned 2940	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ (𝑄‘0))
42	18, 27, 31, 41	leneltd 11294	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) < 𝐶)
43		fveq2 6835	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑄‘𝑘) = (𝑄‘0))
44	43	breq1d 5096	. . . . . . . 8 ⊢ (𝑘 = 0 → ((𝑄‘𝑘) < 𝐶 ↔ (𝑄‘0) < 𝐶))
45	44	elrab 3635	. . . . . . 7 ⊢ (0 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ↔ (0 ∈ (0..^𝑀) ∧ (𝑄‘0) < 𝐶))
46	14, 42, 45	sylanbrc 584	. . . . . 6 ⊢ (𝜑 → 0 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶})
47	46	ne0d 4283	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ≠ ∅)
48		fzossfz 13627	. . . . . . . 8 ⊢ (0..^𝑀) ⊆ (0...𝑀)
49		fzssz 13474	. . . . . . . . 9 ⊢ (0...𝑀) ⊆ ℤ
50		zssre 12525	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ
51	49, 50	sstri 3932	. . . . . . . 8 ⊢ (0...𝑀) ⊆ ℝ
52	48, 51	sstri 3932	. . . . . . 7 ⊢ (0..^𝑀) ⊆ ℝ
53	2, 52	sstri 3932	. . . . . 6 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℝ
54	53	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℝ)
55		fisupcl 9377	. . . . 5 ⊢ (( < Or ℝ ∧ ({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ∈ Fin ∧ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ≠ ∅ ∧ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℝ)) → sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶})
56	4, 8, 47, 54, 55	syl13anc 1375	. . . 4 ⊢ (𝜑 → sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶})
57	2, 56	sselid 3920	. . 3 ⊢ (𝜑 → sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ∈ (0..^𝑀))
58	1, 57	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀))
59	48, 58	sselid 3920	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
60	15, 59	ffvelcdmd 7032	. . . 4 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ)
61	60	rexrd 11189	. . 3 ⊢ (𝜑 → (𝑄‘𝐼) ∈ ℝ^*)
62		fzofzp1 13713	. . . . . 6 ⊢ (𝐼 ∈ (0..^𝑀) → (𝐼 + 1) ∈ (0...𝑀))
63	58, 62	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 + 1) ∈ (0...𝑀))
64	15, 63	ffvelcdmd 7032	. . . 4 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ)
65	64	rexrd 11189	. . 3 ⊢ (𝜑 → (𝑄‘(𝐼 + 1)) ∈ ℝ^*)
66	1, 56	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶})
67		fveq2 6835	. . . . . . 7 ⊢ (𝑘 = 𝐼 → (𝑄‘𝑘) = (𝑄‘𝐼))
68	67	breq1d 5096	. . . . . 6 ⊢ (𝑘 = 𝐼 → ((𝑄‘𝑘) < 𝐶 ↔ (𝑄‘𝐼) < 𝐶))
69	68	elrab 3635	. . . . 5 ⊢ (𝐼 ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ↔ (𝐼 ∈ (0..^𝑀) ∧ (𝑄‘𝐼) < 𝐶))
70	66, 69	sylib 218	. . . 4 ⊢ (𝜑 → (𝐼 ∈ (0..^𝑀) ∧ (𝑄‘𝐼) < 𝐶))
71	70	simprd 495	. . 3 ⊢ (𝜑 → (𝑄‘𝐼) < 𝐶)
72	52, 58	sselid 3920	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℝ)
73		ltp1 11989	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℝ → 𝐼 < (𝐼 + 1))
74		id 22	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℝ → 𝐼 ∈ ℝ)
75		peano2re 11313	. . . . . . . . . . 11 ⊢ (𝐼 ∈ ℝ → (𝐼 + 1) ∈ ℝ)
76	74, 75	ltnled 11287	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℝ → (𝐼 < (𝐼 + 1) ↔ ¬ (𝐼 + 1) ≤ 𝐼))
77	73, 76	mpbid 232	. . . . . . . . 9 ⊢ (𝐼 ∈ ℝ → ¬ (𝐼 + 1) ≤ 𝐼)
78	72, 77	syl 17	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐼 + 1) ≤ 𝐼)
79	48, 49	sstri 3932	. . . . . . . . . . . 12 ⊢ (0..^𝑀) ⊆ ℤ
80	2, 79	sstri 3932	. . . . . . . . . . 11 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℤ
81	80	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℤ)
82		elrabi 3631	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} → ℎ ∈ (0..^𝑀))
83		elfzo0le 13652	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ (0..^𝑀) → ℎ ≤ 𝑀)
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} → ℎ ≤ 𝑀)
85	84	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) → ℎ ≤ 𝑀)
86	85	ralrimiva 3130	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑀)
87		breq2 5090	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑀 → (ℎ ≤ 𝑚 ↔ ℎ ≤ 𝑀))
88	87	ralbidv 3161	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → (∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚 ↔ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑀))
89	88	rspcev 3565	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑀) → ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚)
90	11, 86, 89	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚)
91	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚)
92		elfzuz 13468	. . . . . . . . . . . . . 14 ⊢ ((𝐼 + 1) ∈ (0...𝑀) → (𝐼 + 1) ∈ (ℤ_≥‘0))
93	63, 92	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐼 + 1) ∈ (ℤ_≥‘0))
94	93	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ (ℤ_≥‘0))
95	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝑀 ∈ ℤ)
96	51, 63	sselid 3920	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ)
97	96	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ ℝ)
98	95	zred 12627	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝑀 ∈ ℝ)
99		elfzle2 13476	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 + 1) ∈ (0...𝑀) → (𝐼 + 1) ≤ 𝑀)
100	63, 99	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝑀)
101	100	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ≤ 𝑀)
102		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝑄‘(𝐼 + 1)) < 𝐶)
103	64	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝑄‘(𝐼 + 1)) ∈ ℝ)
104	27	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝐶 ∈ ℝ)
105	103, 104	ltnled 11287	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ((𝑄‘(𝐼 + 1)) < 𝐶 ↔ ¬ 𝐶 ≤ (𝑄‘(𝐼 + 1))))
106	102, 105	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ¬ 𝐶 ≤ (𝑄‘(𝐼 + 1)))
107		iccleub 13348	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑄‘0) ∈ ℝ^* ∧ (𝑄‘𝑀) ∈ ℝ^* ∧ 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) → 𝐶 ≤ (𝑄‘𝑀))
108	28, 29, 26, 107	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ≤ (𝑄‘𝑀))
109	108	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑀 = (𝐼 + 1)) → 𝐶 ≤ (𝑄‘𝑀))
110		fveq2 6835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 = (𝐼 + 1) → (𝑄‘𝑀) = (𝑄‘(𝐼 + 1)))
111	110	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑀 = (𝐼 + 1)) → (𝑄‘𝑀) = (𝑄‘(𝐼 + 1)))
112	109, 111	breqtrd 5112	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑀 = (𝐼 + 1)) → 𝐶 ≤ (𝑄‘(𝐼 + 1)))
113	112	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) ∧ 𝑀 = (𝐼 + 1)) → 𝐶 ≤ (𝑄‘(𝐼 + 1)))
114	106, 113	mtand 816	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → ¬ 𝑀 = (𝐼 + 1))
115	114	neqned 2940	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → 𝑀 ≠ (𝐼 + 1))
116	97, 98, 101, 115	leneltd 11294	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) < 𝑀)
117		elfzo2 13610	. . . . . . . . . . . 12 ⊢ ((𝐼 + 1) ∈ (0..^𝑀) ↔ ((𝐼 + 1) ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ (𝐼 + 1) < 𝑀))
118	94, 95, 116, 117	syl3anbrc 1345	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ (0..^𝑀))
119		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐼 + 1) → (𝑄‘𝑘) = (𝑄‘(𝐼 + 1)))
120	119	breq1d 5096	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐼 + 1) → ((𝑄‘𝑘) < 𝐶 ↔ (𝑄‘(𝐼 + 1)) < 𝐶))
121	120	elrab 3635	. . . . . . . . . . 11 ⊢ ((𝐼 + 1) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ↔ ((𝐼 + 1) ∈ (0..^𝑀) ∧ (𝑄‘(𝐼 + 1)) < 𝐶))
122	118, 102, 121	sylanbrc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶})
123		suprzub 12883	. . . . . . . . . 10 ⊢ (({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶} ⊆ ℤ ∧ ∃𝑚 ∈ ℤ ∀ℎ ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}ℎ ≤ 𝑚 ∧ (𝐼 + 1) ∈ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}) → (𝐼 + 1) ≤ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ))
124	81, 91, 122, 123	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ≤ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ))
125	124, 1	breqtrrdi 5128	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) < 𝐶) → (𝐼 + 1) ≤ 𝐼)
126	78, 125	mtand 816	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑄‘(𝐼 + 1)) < 𝐶)
127		eqcom 2744	. . . . . . . . . . 11 ⊢ ((𝑄‘(𝐼 + 1)) = 𝐶 ↔ 𝐶 = (𝑄‘(𝐼 + 1)))
128	127	biimpi 216	. . . . . . . . . 10 ⊢ ((𝑄‘(𝐼 + 1)) = 𝐶 → 𝐶 = (𝑄‘(𝐼 + 1)))
129	128	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → 𝐶 = (𝑄‘(𝐼 + 1)))
130	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → 𝑄 Fn (0...𝑀))
131	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → (𝐼 + 1) ∈ (0...𝑀))
132		fnfvelrn 7027	. . . . . . . . . 10 ⊢ ((𝑄 Fn (0...𝑀) ∧ (𝐼 + 1) ∈ (0...𝑀)) → (𝑄‘(𝐼 + 1)) ∈ ran 𝑄)
133	130, 131, 132	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → (𝑄‘(𝐼 + 1)) ∈ ran 𝑄)
134	129, 133	eqeltrd 2837	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘(𝐼 + 1)) = 𝐶) → 𝐶 ∈ ran 𝑄)
135	32, 134	mtand 816	. . . . . . 7 ⊢ (𝜑 → ¬ (𝑄‘(𝐼 + 1)) = 𝐶)
136	126, 135	jca 511	. . . . . 6 ⊢ (𝜑 → (¬ (𝑄‘(𝐼 + 1)) < 𝐶 ∧ ¬ (𝑄‘(𝐼 + 1)) = 𝐶))
137		pm4.56 991	. . . . . 6 ⊢ ((¬ (𝑄‘(𝐼 + 1)) < 𝐶 ∧ ¬ (𝑄‘(𝐼 + 1)) = 𝐶) ↔ ¬ ((𝑄‘(𝐼 + 1)) < 𝐶 ∨ (𝑄‘(𝐼 + 1)) = 𝐶))
138	136, 137	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ ((𝑄‘(𝐼 + 1)) < 𝐶 ∨ (𝑄‘(𝐼 + 1)) = 𝐶))
139	64, 27	leloed 11283	. . . . 5 ⊢ (𝜑 → ((𝑄‘(𝐼 + 1)) ≤ 𝐶 ↔ ((𝑄‘(𝐼 + 1)) < 𝐶 ∨ (𝑄‘(𝐼 + 1)) = 𝐶)))
140	138, 139	mtbird 325	. . . 4 ⊢ (𝜑 → ¬ (𝑄‘(𝐼 + 1)) ≤ 𝐶)
141	27, 64	ltnled 11287	. . . 4 ⊢ (𝜑 → (𝐶 < (𝑄‘(𝐼 + 1)) ↔ ¬ (𝑄‘(𝐼 + 1)) ≤ 𝐶))
142	140, 141	mpbird 257	. . 3 ⊢ (𝜑 → 𝐶 < (𝑄‘(𝐼 + 1)))
143	61, 65, 27, 71, 142	eliood 45949	. 2 ⊢ (𝜑 → 𝐶 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
144		fveq2 6835	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑄‘𝑗) = (𝑄‘𝐼))
145		oveq1 7368	. . . . . 6 ⊢ (𝑗 = 𝐼 → (𝑗 + 1) = (𝐼 + 1))
146	145	fveq2d 6839	. . . . 5 ⊢ (𝑗 = 𝐼 → (𝑄‘(𝑗 + 1)) = (𝑄‘(𝐼 + 1)))
147	144, 146	oveq12d 7379	. . . 4 ⊢ (𝑗 = 𝐼 → ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))) = ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))))
148	147	eleq2d 2823	. . 3 ⊢ (𝑗 = 𝐼 → (𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))) ↔ 𝐶 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))))
149	148	rspcev 3565	. 2 ⊢ ((𝐼 ∈ (0..^𝑀) ∧ 𝐶 ∈ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))))
150	58, 143, 149	syl2anc 585	1 ⊢ (𝜑 → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 ⊆ wss 3890 ∅c0 4274 class class class wbr 5086 Or wor 5532 ran crn 5626 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 Fincfn 8887 supcsup 9347 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 ℝ^*cxr 11172 < clt 11173 ≤ cle 11174 ℕcn 12168 ℕ₀cn0 12431 ℤcz 12518 ℤ_≥cuz 12782 (,)cioo 13292 [,]cicc 13295 ...cfz 13455 ..^cfzo 13602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-n0 12432 df-z 12519 df-uz 12783 df-ioo 13296 df-icc 13299 df-fz 13456 df-fzo 13603

This theorem is referenced by: fourierdlem41 46597 fourierdlem48 46603 fourierdlem49 46604 fourierdlem70 46625 fourierdlem71 46626 fourierdlem103 46658 fourierdlem104 46659

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